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Recurrence Solution Techniques in Data Structures: The Substitution, Iteration, and Master Methods

Learn how to apply the substitution, iteration, and master methods to solve recurrence relations in data structures. Practice finding tight asymptotic relations, proving solutions, and analyzing algorithms for Fibonacci series computation.

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Recurrence Solution Techniques in Data Structures: The Substitution, Iteration, and Master Methods

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  1. Foundations of Data Structures Practical Session #3 Recurrence Amihai Savir & Ilya Mirsky - 2013

  2. Solution Methods • Substitution method • Guess the form of the solution and prove it by induction. • Iteration method • Convert the recurrence into a summation and solve it • Mastermethod • Next practical session…

  3. Question 0- warm up • Find the most tight asymptotic relation between the following pairs: • )

  4. Question 1 Prove that using the substitution method. Solution The form of the solution is already given, so we don't have to guess it. We only need to prove that, i.e., that .

  5. Question 1 cont’d

  6. Question 2

  7. Question 3 fact(n) { if (n == 0) return 1 return n * fact(n – 1) }

  8. Question 4 • Fibonacci series is defined as follows: • Find an iterative algorithm and a recursive one for computing element number in Fibonacci series, i.e., Fibonacci(n).Analyze the running-time of each of the algorithms.

  9. Question 4 cont’d recFib(n) { if (n ≤ 1) return n else return recFib(n-1) + recFib(n-2) }

  10. Question 4 cont’d • In the same manner: • The series ends when • To summarize:

  11. Question 4 cont’d iterFib(n) { allocate f[0..n] f[0] = 0 f[1] = 1 for (i=2 ; i ≤ n ; i++) f[i] = f[i-1] + f[i-2] return f[n] }

  12. Question 5 • publicstaticintfindMax(int[]arr) } • returnFindMax(arr,0, arr.length-1); • { • public staticintfindMax(inta[],intleft, intright)} • intmiddle; • intmax_l, max_r; • if( left == right ) • returna[left]; • else } • middle = (left + right) / 2; • max_l= FindMax( a, left, middle); • max_r= FindMax( a, middle+1, right); • returnMath.max(max_l,max_r); • { • {

  13. Question 5 cont’d

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